Monday, July 23, 2012

The Monty Hall problem is named after it's similarity to some of the games that were played on the original Let's Make a Deal television show hosted by Monty Hall. Imagine a room with three doors. Behind two of the doors are goats and the third is a car. You are asked to pick one of the doors. Lets say you choose door #1. The host, who knows what is behind each door then opens one of the other two doors which has a goat behind it. In this case, lets say he opens door #3. At this point the host then gives you an option. Do you want to stay with door #1 or would you like to switch to door #2. The question is, should you stick with your first choice, switch or does it even matter?

Most people (my self included) initially say that it doesn't matter one way or the other because intuitively it seems as if you have a 50/50 chance at getting the car whether you switch or not. In reality, it is always advantageous to switch. The easiest way to understand this is to break down it's probabilities. When you originally chose door #1 there was a 1/3 chance of getting the car and a 2/3 chance of not getting the car. Another way of saying this is that you have a 1/3 chance that the car is behind door #1 and a 2/3 chance that the car is behind door #2 or #3. When the host opens door #3 to reveal the goat, you still have a 1/3 chance of the car being behind door #1 (not switching) but now a 2/3 chance that the car is behind door #2 (switching).

Here is a good video which explains it well

The story behind the problem
Another interesting part of this problem is the story behind it's rise to fame. In her September 1990 column in Parade magazine, Marilyn vos Savant answered the following question posed by a reader:

"Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. He says to you, "Do you want to pick door #2?" Is it to your advantage to switch your choice of doors?"
Craig F. Whitaker
Columbia, Maryland

"Yes; you should switch. The first door has a 1/3 chance of winning, but the second door has a 2/3 chance. Here’s a good way to visualize what happened. Suppose there are a million doors, and you pick door #1. Then the host, who knows what’s behind the doors and will always avoid the one with the prize, opens them all except door #777,777. You’d switch to that door pretty fast, wouldn’t you?"

Many of the response letters that poured in disputing her answer were arrogant and pompous. Often, the worst of these were from academics. Here are a few examples:

"You blew it, and you blew it big! Since you seem to have difficulty grasping the basic principle at work here, I’ll explain. After the host reveals a goat, you now have a one-in-two chance of being correct. Whether you change your selection or not, the odds are the same. There is enough mathematical illiteracy in this country, and we don’t need the world’s highest IQ propagating more. Shame!"

Scott Smith, Ph.D

University of Florida

"Since you seem to enjoy coming straight to the point, I’ll do the same. You blew it! Let me explain. If one door is shown to be a loser, that information changes the probability of either remaining choice, neither of which has any reason to be more likely, to 1/2. As a professional mathematician, I’m very concerned with the general public’s lack of mathematical skills. Please help by confessing your error and in the future being more careful."

Robert Sachs, Ph.D.

George Mason University

Marilyn wrote another column providing more support to her answer but the negative responses kept coming:

"May I suggest that you obtain and refer to a standard textbook on probability before you try to answer a question of this type again?"

Charles Reid, Ph.D.
University of Florida

"I am sure you will receive many letters on this topic from high school and college students. Perhaps you should keep a few addresses for help with future columns."

W. Robert Smith, Ph.D.
Georgia State University

"You are utterly incorrect about the game show question, and I hope this controversy will call some public attention to the serious national crisis in mathematical education. If you can admit your error, you will have contributed constructively towards the solution of a deplorable situation. How many irate mathematicians are needed to get you to change your mind?"

E. Ray Bobo, Ph.D.

Georgetown University

But then, after some time had passed, the letters started to turn in favor of support:

"You are indeed correct. My colleagues at work had a ball with this problem, and I dare say that most of them, including me at first, thought you were wrong!"

Seth Kalson, Ph.D

Massachusetts Institute of Technology

"The teachers in my graduate-level mathematics classes, most of whom thought you were wrong, conducted your experiment as a class project. Each of the twenty-five teachers had students in their middle or high school classes play at least 400 games. In all, we had 14,800 samples of the experiment, and we’re convinced that you were correct —the contestant should switch!"

Eloise Rudy, Furman University
Greenville, South Carolina

In the end, after countless simulations were run, Marilyn vos Savant was proven right. I'm sure her sense of feeling vindicated must have been quite sweet.

Tomorrow The Dark Knight Rises opens. It is expected to be a very dark film, so I thought I would close out my series on Batman fan films with something on the light side. From College Humor, here is Batman Chooses His Voice.

Tuesday, July 17, 2012

Only a few days away from The Dark Knight Rises, just enough time to squeeze in a couple more Batman fan films. World's Finest is a fan film created in 2004 by Sandy Collora (see Batman: Dead End). Like the fan film Grayson, World's Finest is a trailer to a movie that does not exist. This format works well for fan films, allowing directors with little to no budget to pay homage to the characters in the span of a few minutes. The ambiguity of the trailer format leaves the viewer to use their imagination to fill in the gaps, which is something most comic book readers are used to.

Monday, July 16, 2012

Korean artist Seung Mo Park created these stunning images by cutting layer after layer of wire mesh. The use of these unique materials and methods results in these mesmerizing pieces which seem to float in the air like phantasms. See the video at the bottom which demonstrates the complete process.

Friday, July 13, 2012

Conventional wisdom says that investing in stocks in the short run, say in a period of 5 or less years, is very risky due to volatility in the markets. Extend the time frame to 15 or more years, the risk is minimal and they are the superior investment. Supporting evidence for such claims often include this chart of the Dow Jones Industrial Average from 1900 to the present.

Dow Jones Industrial Average - Click to Enlarge

It is obvious that stocks performed amazingly well after the great depression into the start of the 21st century. Unfortunately over the last 10 years things really haven't been going so well. If we are to follow conventional wisdom then this really isn't anything to worry about since short term volatility is supposed to be part of the game. But should we really be so secure with the idea that everything will turn out alright if we just wait long enough?

If looking at the historical performance of the Dow Jones is supporting evidence for stocks being safe in the long run, then looking at the historical performance of stocks in other developed nations seems relevant. With this in mind, here is a chart of Japan's Nikkei 225 index from 01/4/84 to 06/01/12.

Nikkei 225 Index - Click to Enlarge

As you can see, it has been over two decades since Japan's stock bubble burst and prices are essentially where they were back in 1984. So if back in the early 1980's you were a young Japanese business man saving for retirement investing heavily in stocks, and you continued to do so to the present day, you would have taken a big lose on your investment.

My point here is not to suggest that people avoid investing in the stock market. I am only trying to show that there is nothing absolute about the common idea that stocks are a safe bet in the long run. The truth of the matter is that there is a tremendous amount of uncertainty in the areas of economics and politics; much too much to naively believe in this comfy bed time story.

Monday, July 9, 2012

Eleven more days till the opening of The Dark Knight Rises and I'm continuing my count down with another fan film short with Batman vs Wolverine. Lets see how the caped crusader stacks up against Weapon X this time.

Sunday, July 1, 2012

Seeds of Arkham is the 2011 sequel to Batman: City of Scars. Bat in the Sun productions consistently puts out quality fan films and this is no exception. It is non-stop action from beginning to end. Two thumbs up in my opinion (yes I can vote twice, I have two thumbs).